What is the Domain of the Function y = sqrt(x^2 - 1) for Finding Critical Point?

In summary, the conversation discussed a problem involving finding a critical point of a function based on its derivative. The definition of a critical point was also mentioned, stating that it is an interior point of the domain where the derivative is either zero or undefined. The problem involved finding the critical point of the function y = sqrt(x^2 - 1) and it was determined that the critical point is at x = 0. It was also noted that the function is undefined at x = 0, satisfying both conditions for a critical point. The conversation also touched on the domain of the function, with it being mentioned that the values of -1 and 1 are not critical points because they do not exist in the real domain of the function.
  • #1
donjt81
71
0
Ok so I am trying to do this problem and I have a question

So based on the definition given in the book "An interior point of the domain of a function f where f' is zero or undefined is a critical point of f"

This is the problem:
y = sqrt(x^2 - 1)
so
y' = x/sqrt(x^2 - 1)

to find a critical point
y' = 0
x/sqrt(x^2 - 1) = 0
x = 0

also to find the critical point we have to see if y' will be undefined at any value of x. as we can see y' will be undefined at x = 0.

so from the first condition when we solved for y' = 0, we got x = 0 and now for the second condition y' is undefined at x = 0.

So both conditions are satisfied at x = 0 so does that mean the critical point is at x = 0. In the definition it says first condition or second condition has to be satisfied. I might be reading too much into the definition.
 
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  • #2
anyone...?
 
  • #3
if x = -1, or 1 then f' is undefined. 1 and -1 also exist in f(x).

-1 < x < 1 do not exist (real) in f(x) so they are not critical points
when x = 0 (included in the inequality above) f(x) does not exist.

Notice f(x=0) means, sqrt(0^2-1) = sqrt(-1) = i
 
  • #4
What IS the domain of that function?
 

1. What is a critical point?

A critical point is a point on a graph where the derivative is equal to zero. This means that at this point, the function is not increasing or decreasing, and it can be a maximum, minimum, or an inflection point.

2. How do you find the critical points of a function?

To find the critical points of a function, you need to take the derivative of the function and set it equal to zero. Then, solve for the variable to find the x-values of the critical points. You can also use the second derivative test to determine if a critical point is a maximum, minimum, or inflection point.

3. Why are critical points important?

Critical points are important because they give information about the behavior of a function. They can help determine the maximum and minimum values of a function, as well as any points of inflection. They are also used in optimization problems to find the optimal solution.

4. Can a function have more than one critical point?

Yes, a function can have multiple critical points. This can happen when the function has multiple local extrema or points of inflection. It is important to consider all critical points when analyzing the behavior of a function.

5. What is the difference between a critical point and a singular point?

A critical point is a point on a graph where the derivative is equal to zero, while a singular point is a point where the function is not defined or the derivative is undefined. Critical points can occur at singular points, but not all singular points are critical points.

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